Proof Of Algorithm Research Articles

An Algorithmic Approach to the q-Summability Problem of Bivariate Rational Functions

In 2014, Chen and Singer solved the summability problem of bivariate rational functions. Later an algorithmic proof was presented by Hou and the author. In this paper, the algorithm will be simplified and adapted to the q-case.

Integration of Internet of Things and blockchain toward portability and low‐energy consumption

AbstractThe modern era of information and technology leverages connectivity and information sharing for enhancing applications and reducing human intervention. The proliferation of Internet of Things (IoT) in almost all aspects of modern life raises major challenges of security and privacy in health care services and the pharmaceutical industry. Blockchain has emerged as a secure technology in a trustless network without central governance. The inherent attributes it provides can solve the security and privacy challenges related to IoT and extend its capabilities. However, traditional instances of Blockchain are not suited for the IoT environment for reasons such as computationally expensive consensus process. Proof of authentication is a lightweight consensus algorithm that can be implemented in the IoT environment. This paper presents an IoT‐friendly Blockchain scheme implemented to evaluate the feasibility of medical supply and drug transportation to aid security and mitigate privacy issues which are fault‐tolerant, transparent, and traceable. The proof of concept implementation on portable single board computer demonstrates proof of authentication consensus algorithm executed with block validation and block addition in 30 and 40 ms while consuming 45 and 60mJ, respectively. Additionally, the network performance of the proposed architecture was presented and evaluated in the application context.

Blockchain-based Perfect Sharing Project Platform based on the Proof of Atomicity Consensus Algorithm

Blockchain-based Perfect Sharing Project Platform based on the Proof of Atomicity Consensus Algorithm

A New ECDLP-Based PoW Model

Blockchain technology has attracted a lot of research interest in the last few years. Originally, their consensus algorithm was Hashcash, which is an instance of the so-called Proof-of-Work. Nowadays, there are several competing consensus algorithms, not necessarily PoW. In this paper, we propose an alternative proof of work algorithm which is based on the solution of consecutive discrete logarithm problems over the point group of elliptic curves. At the same time, we sketch a blockchain scheme, whose consensus is reached via our algorithm. In the considered architecture, the curves are pseudorandomly determined by block creators, chosen to be cryptographically secure and changed every epoch. Given the current state of the chain and a prescribed set of transactions, the curve selection is fully rigid, therefore trust is needed neither in miners nor in the scheme proposers.

Preventing the generation of inconsistent sets of crisp classification rules

In recent years, the interest in interpretable classification models has grown. One of the proposed ways to improve the interpretability of a classification model based on collections of crisp rules is to use sets (unordered collections) instead of lists (ordered collections). One of the problems associated with sets is that multiple rules may cover a single instance but predict different classes for it, thus requiring a conflict resolution strategy. In this work, we propose two algorithms capable of finding feature-space regions inside which any created rule would be consistent with the already existing rules, preventing inconsistencies from arising. Our algorithms, named CFSGS and CFSBE, do not generate classification rules nor classification models but are instead meant to enhance algorithms that do so, such as Learning Classifier Systems. We analyzed both algorithms from a theoretical perspective and conducted experiments with a proof of concept evolutionary algorithm that employs CFSBE. The experiments suggest that using CFSBE as an embedded tool does incur a computational overhead, but such cost is not prohibitive.

A unified existence theorem for normal spanning trees

We show that a graph G has a normal spanning tree if and only if its vertex set is the union of countably many sets each separated from any subdivided infinite clique in G by a finite set of vertices. This proves a conjecture by Brochet and Diestel from 1994, giving a common strengthening of two classical normal spanning tree criterions due to Jung and Halin.Moreover, our method gives a new, algorithmic proof of Halin's theorem that every connected graph not containing a subdivision of a countable clique has a normal spanning tree.

A Mechanized Proof of a Textbook Type Unification Algorithm

Unification is the core of type inference algorithms for modern functional programming languages, like Haskell and SML. As a first step towards a formalization of a type inference algorithm for such programming languages, we present a formalization in Coq of a type unification algorithm that follows classic algorithms presented in programming language textbooks. We also report on the use of such formalization to build a correct type inference algorithm for the simply typed λ-calculus.

ENERGY CONSUMPTION OF BITCOIN MINING

Blockchain is one of the most popular terms associated with changes in the technological paradigm-taking place within the framework of the so-called fourth industrial revolution. The Proof of Work algorithm is used for process transactions and ensure security in the Bitcoin network. The paper aims energy consumption of bitcoin mining. It implies the need for a network participant to solve a certain cryptographic task with energy consumption and predetermined final time for its completion. The methodology includes the analysis of relationship between active, reactive and apparent power is determined by the phase angle between the current and voltage in the network, more precisely, with the cosine of this angle - cos? (power factor news). A correctly solved task is accepted by the network and included in the public transaction register. The participant who first provided such a solution receives a block reward. The main finding is that the solution of the energy consumption problem by other network participants, an order of magnitude less time is required.

Local Mutual Exclusion algorithm using fuzzy logic for Flying Ad hoc Networks

The Local Mutual Exclusion (LME) problem is a variant of classical Mutual Exclusion (ME) problem and can be considered as an extension of dining philosopher problem. In LME, no two neighboring nodes can enter the critical section (CS) simultaneously, whereas two non-neighboring nodes can be in their CS simultaneously. The resource allocation problem in Flying Ad hoc Networks (FANETs), is relatively an unexplored area despite having several potential applications. The present paper proposes LME problem for FANETs and provides a leader-based algorithm named as Request Collector Local Mutual Exclusion (RCLME) for the same. To the best of our information, LME problem is introduced first time in Flying Ad hoc Networks. The striking feature of the proposed algorithm is the introduction of a fuzzy logic-based leader election that considers the node speed, node direction, link quality, and the distance from the resource. The correctness proof of the RCLME algorithm has been presented. The simulation results show that RCLME algorithm significantly outperforms other related algorithms available in the literature; specially, when the number of nodes is large. The use of fuzzy logic and request collector improves the efficiency, fault tolerating capacity and ability to handle volatility.

A Why3 proof of GMP algorithms

Large-integer arithmetic algorithms are used in contexts where both their performance and their correctness are critical, such as cryptographic software. The fastest algorithms are complex enough that formally verifying them is desirable but challenging. We have formally verified a comprehensive arbitrary-precision integer arithmetic library that implements many state-of-the-art algorithms from the GMP library. The algorithms we have verified include addition, subtraction, Toom-Cook multiplication, division and square root. We use the Why3 platform to perform the proof semi-automatically. We obtain an efficient and formally verified C library of low-level functions on arbitrary-precision natural integers. This verification covers the functional correctness of the algorithms, as well as the memory safety and absence of arithmetic overflows of their C implementation. Using detailed pseudocode, we present the algorithms that we verified and outline their proofs.

Packing Nearly Optimal Ramsey R(3,t) Graphs

In 1995 Kim famously proved the Ramsey bound R(3,t) \ge c t^2/\log t by constructing an n-vertex graph that is triangle-free and has independence number at most C \sqrt{n \log n}. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph K_n into a packing of such nearly optimal Ramsey R(3,t) graphs. More precisely, for any \epsilon>0 we find an edge-disjoint collection (G_i)_i of n-vertex graphs G_i \subseteq K_n such that (a) each G_i is triangle-free and has independence number at most C_\epsilon \sqrt{n \log n}, and (b) the union of all the G_i contains at least (1-\epsilon)\binom{n}{2} edges. Our algorithmic proof proceeds by sequentially choosing the graphs G_i via a semi-random (i.e., Rodl nibble type) variation of the triangle-free process. As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo (concerning a Ramsey-type parameter introduced by Burr, Erdos, Lovasz in 1976). Namely, denoting by s_r(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H=K_3 and establish that s_r(K_3) = \Theta(r^2 \log r).

Two Algorithms for Constructing Independent Spanning Trees in (n,k)-Star Graphs

In a graph $G$ , two spanning trees $T_{1}$ and $T_{2}$ are rooted at the same vertex $r$ . If, for every $v \in V(G)$ , the paths from $v$ to the root $r$ in $T_{1}$ and $T_{2}$ are internally vertex-disjoint, they are called independent spanning trees (ISTs). ISTs can be applied in numerous fields, such as fault-tolerant broadcasting and secure message distribution. The ( $n$ , $k$ )-star graphs $S_{n,k}$ constitute a generalized version of the star network. The method of constructing ISTs in ( $n$ , $k$ )-star graphs remains unknown. In this paper, we propose one recursive algorithm and one parallel algorithm for constructing ISTs on ( $n$ , $k$ )-star graphs. The main ideas of the recursive algorithm are to use induction to change small trees into large trees, to use a modified breadth-first search (MBFS) traversal to create a frame for an IST, and to use a breadth-first search (BFS) traversal to connect the rest of nodes. The main ideas of the parallel algorithm are to create frames through MBFS traversals in parallel, and to use some specific rules to connect the rest of nodes in parallel. We also present validation proofs for both algorithms, and analyze the time complexities of both algorithms. The time complexity of the recursive algorithm in $S_{n,k}$ is $O\left({n\times \frac {n!}{(n-k)!}}\right)$ , where $\frac {n!}{(n-k)!}$ is the number of nodes of $S_{n,k}$ . The time complexity of the parallel algorithm can be reduced to $O\left({\frac {n!}{(n-k)!}}\right)$ if the system has $n-1$ processors computing in parallel. Both algorithms are correct with the stated time complexity values; the parallel algorithm is more efficient than the recursive algorithm.

Online Payment Using Blockchain

Increasing list of records is with blockchain where each record is linked with the help of cryptography. Every block in the chain contains timestamp, transaction details and hash of a previous block, hash is cryptographic hash. This is a secured system, which we plan to replace the current online payment system. A current online payment gateway is prone to hackers where the attacker can tamper into the network, thus creating money loss. And not only this but also the transaction has to go through multiple payment systems which consumes time, also creating a risk of transaction getting failed. So, our system would be using blockchain that allows online transactions which would allow online payments to be sent directly from one party to another without going through a financial institution and in a secured way. This system allows online transactions between two parties based on cryptographic proof without relying and trusting for a third party. To record transactions, we use proof of work algorithm which makes computationally impractical for an attacker to change. Digital signatures provide part of the solution for ensuring the security and integrity of the data that is recorded onto a blockchain.

MEMS-Based IMU Drift Minimization: Sage Husa Adaptive Robust Kalman Filtering

The Attitude Heading Reference System (AHRS) has been widely used to provide the position and orientation of a rigid body. A low cost MEMS based inertial sensor measurement unit (IMU) is a core device in AHRS. To improve the AHRS system performance, there is a need to develop (i) stochastic IMU error models and (ii) random noise minimization techniques. In this paper, we modify the Sage-Husa Adaptive Kalman Filter (SHAKF) to incorporate time-varying noise estimator and robustifier, termed as Modified Sage-Husa Adaptive Robust Kalman Filter (MSHARKF). In the proposed algorithm, a three segment approach is used to evaluate the adaptive scale factor followed by the learning statistics. The scale factor is iteratively updated in the MSHARKF equations. In addition, angle random walk (ARW) and bias instability (BI) errors are represented by state-space models. The proposed algorithm is applied to restrain the drift error and random noise in the MEMS IMUs signals. The performance of this algorithm is analyzed using Allan variance (AV) analysis for static signals whereas the Root Mean Square Error (RMSE) values are evaluated for dynamic signals. Experimental results demonstrate the effectiveness of MSHARKF in reducing the drift and random noise in static and dynamic conditions as compared with other existing algorithms. Finally, we present sufficient conditions for convergence proof of the MSHARKF algorithm.

E-Voting: Blockchain Based Voting For Secure Vote Casting

blockchain is decentralized immutable ledger technology maintaining integrity. So to conduct tamperproof election it’s one of the approach towards it. Smart contracts are Self executed code that is written on Ethereum platform in blockchain. An E-voting system should be completely secure and does not allow voting twice that is double spending in blockchain. So it should be completely transparent. In research work electronic voting application is implemented and tested using smart contract on Ethereum platform with the help of metamask wallet. The results of ballots and votes will be stored on Ethereum blockchain with the help of consensus algorithm proof of stake. This consensus is used in validating a transaction with concept of majority approval. Current electronic voting system requires a centralized authority to control the procedure from ballot input to result output and for monitoring of election. While blockchain technology provide decentralized system which is open across connected nodes. Blockchain assets provide increased level of system security from hacking and fraud. Every transaction in blockchain is time-stamped and signed digitally with the help of cryptographic algorithms, and it assigns unique hash value to every block so it can be trace easily. Blockchain technology is one of solutions because it embraces a decentralized system and the entire databases are owned by many users. The blockchain technology also has much vulnerability due to which many attacks like 51% attack, Double Spending attack, DDOS attack, Sybil attack, Eclipse attack and Routing attack can be performed on it.

Overview of the New Version 3 NASA Micro-Pulse Lidar Network (MPLNET) Automatic Precipitation Detection Algorithm

Precipitation modifies atmospheric column thermodynamics through the process of evaporation and serves as a proxy for latent heat modulation. For this reason, a correct precipitation parameterization (especially for low-intensity precipitation) within global scale models is crucial. In addition to improving our modeling of the hydrological cycle, this will reduce the associated uncertainty of global climate models in correctly forecasting future scenarios, and will enable the application of mitigation strategies. In this manuscript we present a proof of concept algorithm to automatically detect precipitation from lidar measurements obtained from the National Aeronautics and Space Administration Micropulse lidar network (MPLNET). The algorithm, once tested and validated against other remote sensing instruments, will be operationally implemented into the network to deliver a near real time (latency <1.5 h) rain masking variable that will be publicly available on MPLNET website as part of the new Version 3 data products. The methodology, based on an image processing technique, detects only light precipitation events (defined by intensity and duration) such as light rain, drizzle, and virga. During heavy rain events, the lidar signal is completely extinguished after a few meters in the precipitation or it is unusable because of water accumulated on the receiver optics. Results from the algorithm, in addition to filling a gap in light rain, drizzle, and virga detection by radars, are of particular interest for the scientific community as they help to fully characterize the aerosol cycle, from emission to deposition, as precipitation is a crucial meteorological phenomenon accelerating atmospheric aerosol removal through the scavenging effect. Algorithm results will also help the understanding of long term aerosol–cloud interactions, exploiting the multi-year database from several MPLNET permanent observational sites across the globe. The algorithm is also applicable to other lidar and/or ceilometer network infrastructures in the framework of the Global Aerosol Watch (GAW) aerosol lidar observation network (GALION).

Acyclic coloring of graphs and entropy compression method

Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, Dujmović et al. (2016) initiated the use of the so-called entropy compression method for graph coloring problems. Then, using the same approach Esperet and Parreau (2013) proved new upper bounds for several chromatic numbers, and explained how that approach could be used for many different coloring problems. Here, we follow this line of research for the particular case of acyclic coloring: we show that every graph with maximum degree Δ has acyclic chromatic number at most 32Δ43+O(Δ).

Creating a Novel Consensus Algorithm for Distributed Computing use Cases

There are many consensus algorithms that exist in parallel computing that involve multiple computing units like virtual machines which make use of available resources and arrive at a single agreeable state for the combined system. This is done on the basis of voting which itself branches into several arrangements like voting, functions of central tendencies, weighted functions of central tendencies etc. Some applications that consensus algorithms try to cover are: deciding on transaction operations (read, write, commit); deciding on node leaders of a system; maintaining replicas in the state of a machine (also called a state machine) and creating consistency between them. Some common algorithms of this type are Proof of Work algorithm (PoW), the practical Byzantine fault tolerance algorithm (PBFT), the proof-of-stake algorithm (PoS) and the delegated proof-of-stake algorithm (DPoS), Paxos algorithm and the Raft consensus algorithm.

SPARK by Example

This paper presents SPARK by Example [10], a guide for people wanting to get involved in formal verification of SPARK programs. SPARK by Example is inspired by ACSL by Example, a similar effort for C/ACSL programs, and provides detailed specification, implementation and proof of classic algorithms (array manipulation, sorting, heap etc). A comparison between ACSL and SPARK is done in the light of proof performance and ease of use.

Directed Lovász local lemma and Shearer’s lemma

Moser and Tardos (J. ACM (JACM) 57(2), 11 2010) gave an algorithmic proof of the lopsided Lovász local lemma (LLL) in the variable framework, where each of the undesirable events is assumed to depend on a subset of a collection of independent random variables. For the proof, they define a notion of a lopsided dependency between the events suitable for this framework. In this work, we strengthen this notion, defining a novel directed notion of dependency and prove the LLL for the corresponding graph. We show that this graph can be strictly sparser (thus the sufficient condition for the LLL weaker) compared with graphs that correspond to other extant lopsided versions of dependency. Thus, in a sense, we address the problem “find other simple local conditions for the constraints (in the variable framework) that advantageously translate to some abstract lopsided condition” posed by Szegedy (2013). We also give an example where our notion of dependency graph gives better results than the classical Shearer lemma. Finally, we prove Shearer’s lemma for the dependency graph we define. For the proofs, we perform a direct probabilistic analysis that yields an exponentially small upper bound for the probability of the algorithm that searches for the desired assignment to the variables not to return a correct answer within n steps. In contrast, the method of proof that became known as the entropic method, gives an estimate of only the expectation of the number of steps until the algorithm returns a correct answer, unless the probabilities are tinkered with.